Public Mathlib landmark · existing upstream theorem · read-only
ProofAtlas

Linear algebra and analysis · existing Mathlib theorem

Cauchy–Schwarz Inequality

The amount by which two real or complex inner-product-space vectors align can never exceed the product of their lengths.

Four separate status axes

Upstream indexedPinned source bytes verified locally
Locally reproducedExact upstream declaration replayed
Reviewed page2 of 3 presentation reviews recorded
Accepted Atlas resultNot recorded for the preferred artifact

These states distinguish upstream identity, local reproduction, review, and Atlas acceptance. This page is part of the public, read-only Mathlib landmark collection.

Formal orientation

Statement map

Statement map for Cauchy–Schwarz InequalityThe norm of an inner product is at most the product of the two vector norms. The pinned upstream declaration is norm_inner_le_norm. The exact checked statement is theorem norm_inner_le_norm {ᵕ : Type*} {E : Type*} [RCLike ᵕ] [SeminormedAddCommGroup E] [InnerProductSpace ᵕ E] (x y : E) : ‖inner ᵕ x y‖ ≤ ‖x‖ * ‖y‖.Mathematical readingThe norm of an innerproduct is at most theproduct of the twovector norms.Pinned declarationmathlib ·norm_inner_le_normExact checked formtheoremnorm_inner_le_norm {ᵕ :Type*} {E : Type*}[RCLike ᵕ][SeminormedAddCommGroupE] [InnerProductSpace ᵕE] (x y : E) : ‖inner ᵕx y‖ ≤ ‖x‖ * ‖y‖Statement map for Cauchy–Schwarz InequalityThe norm of an inner product is at most the product of the two vector norms. The pinned upstream declaration is norm_inner_le_norm. The exact checked statement is theorem norm_inner_le_norm {ᵕ : Type*} {E : Type*} [RCLike ᵕ] [SeminormedAddCommGroup E] [InnerProductSpace ᵕ E] (x y : E) : ‖inner ᵕ x y‖ ≤ ‖x‖ * ‖y‖.Mathematical readingThe norm of an innerproduct is at most theproduct of the twovector norms.Pinned declarationmathlib ·norm_inner_le_normExact checked formtheoremnorm_inner_le_norm {ᵕ :Type*} {E : Type*}[RCLike ᵕ][SeminormedAddCommGroupE] [InnerProductSpace ᵕE] (x y : E) : ‖inner ᵕx y‖ ≤ ‖x‖ * ‖y‖

This deterministic map orients the reader from the mathematical summary to the pinned declaration and exact checked form. The HTML statement below is authoritative.

The theorem at a glance

Cauchy–Schwarz inequality at a glance

A source-bound editorial overview of the exact Mathlib formulation. The image is explanatory; the statement map, formal type, and pinned source remain authoritative.

Vector alignment is bounded by the product of vector lengths. Generated explanation only; the exact HTML statement and pinned source are authoritative.
Detailed visual description

The poster presents Mathlib's real-or-complex inner-product-space inequality, a projection-based interpretation, and the boundary that this declaration alone does not characterize equality.

The inner-product magnitude cannot exceed the product of the two vector norms. Generated explanation only; the exact HTML statement and pinned source are authoritative.
Detailed visual description

A sparse inner-product-space construction isolates two vectors, the projected component, and a perpendicular remainder without adding an equality condition not present in the selected declaration.

Complementary intuition

A mathematical landmark

Cauchy–Schwarz is the basic inequality of inner-product geometry. It controls projections, yields the triangle inequality, and supports foundational estimates across linear algebra, analysis, probability, and geometry.

The low-text schematic gives the theorem a visual identity; it does not replace the source-bound poster or exact statement.

Authoritative formal type

Exact Mathlib statement

theorem norm_inner_le_norm {ᵕ : Type*} {E : Type*} [RCLike ᵕ] [SeminormedAddCommGroup E] [InnerProductSpace ᵕ E] (x y : E) : ‖inner ᵕ x y‖ ≤ ‖x‖ * ‖y‖
Selected evidence declaration text
theorem norm_inner_le_norm {ᵕ : Type*} {E : Type*} [RCLike ᵕ] [SeminormedAddCommGroup E] [InnerProductSpace ᵕ E] (x y : E) : ‖inner ᵕ x y‖ ≤ ‖x‖ * ‖y‖

Claim boundary

No new theorem is claimed

This target indexes Mathlib's norm inequality for the inner product of two vectors in a real or complex inner-product space. It does not include the equality characterization or claim a new proof.

Provenance

Origin and evidence stay separate

Existing declaration
norm_inner_le_norm in mathlib
Relationship
The checked artifact is the upstream declaration itself
Preferred checked artifact
artifact.library.mathlib.cauchy-schwarz-inequality.v001
Source handling
Verified upstream reference; no mirrored source package